Wave Equations

# Wave Equations - Wave Equations 1D Shear Wave H y(x t x τ...

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Unformatted text preview: Wave Equations 1D Shear Wave H y(x, t) x τ + d τ τ σ h σ h dx ρ Soil layer 2 2 2 2 2 1 t y C x y s ∂ ∂ = ∂ ∂ t i Xe t x y ϖ = ) , ( + = x C B x C A X s s ϖ ϖ cos sin π ϖ - = 2 1 2 i H C s i = x C X s i i ϖ sin For free vibration Assume Apply boundary conditions (i) At bottom (x = 0) X(x=0) = 0 (ii) At top (x = H) X’(x=0) = 0 i = 1, 2, … i = 1, 2, … Mode shapes Natural frequencies For forced vibration ( 29 ) ( 1 1 ) ( ) , ( 1 2 2 2 2 2 2 2 2 2 2 2 t y C t y C x y t y t x y t C x y g s s g s = ∂ ∂- ∂ ∂ ⇒ + ∂ ∂ = ∂ ∂ ∑ ∞ = = 1 ) ( ) ( ) , ( i i i x X t q t x y ∫ = H j i dx x X x X ) ( ) ( Orthogonality of the mode shapes ) ( ) ( ) ( 2 t y t q t q g i i i i α ϖ- = + π α ) 1 2 ( 4- = i i let 1D Torsional Wave 2 2 2 2 2 1 t C x s ∂ ∂ = ∂ ∂ θ θ x θ l B.C. At x = 0 θ (0,t)=0 At x = l θ ’(l,t)=0 l G l C s 2 2 1 π ρ π ϖ = = G can be determined by measuring ϖ 1 in a resonance column test! 1D Longitudinal Wave F F+dF q (force/length) 1 1 2 2 2 2 2 = + ∂ ∂- ∂ ∂ q EA t u C x u l x, u ρ , A d’Alembert’s Solution to 1D Wave Equation ) ( ) ( ) , ( ct x ct x t x y + +- = ψ φ 2 2 2 2 1 t y C x y ∂ ∂ = ∂ ∂ Wave equation: Solutions: ) ( ) ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) , ( ' ' x g t ct x x t ct x x t x y x f x x x y = ∂ + ∂ + ∂- ∂ = ∂ ∂ = + = ψ φ ψ φ 2 1 ) ( 2 1 ) ( 2 1 ) ( ) ( 2 1 ) ( 2 1 ) ( k ds s g C x f x k ds s g C x f x x x + + = +- = ∫ ∫ ψ ϕ ( 29 [ ] ∫ +- + + +- = ct x ct x ds s g ct x f ct x f t x y ) ( 2 1 ) ( 2 1 ) , ( Initial conditions: Non-uniform Beam...
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Wave Equations - Wave Equations 1D Shear Wave H y(x t x τ...

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